This example uses a deforming grid approach to simulate aeroelastic
motion of a modified F-15 fighter jet configuration known as NASA research
aircraft #837, shown in Fig. 1. The
computational model assumes half-plane symmetry in the spanwise direction. The
grid consists of 4,715,852 nodes and 27,344,343 tetrahedral elements and
includes detailed features of the external airframe as well as the internal
ducting upstream of the engine fan face and plenum/nozzle combination
downstream of the turbine. For the current test, the freestream Mach number
is 0.90, the angle of attack is 0 degrees, and the Reynolds number based on
the MAC is 1 million. The static pressure ratio at the engine fan face is set
to 0.9 and the total pressure ratio at the plenum face is ramped linearly from
1.0 to its final value of 5.0 over the first 50 time steps.

Figure 1. Modified F-15 with engine duct geometry.

The prescribed grid motion consists of 5 Hz 0.3 degree oscillatory rotations
of the canard, wing, and tail surfaces about their root chordlines, with the
wing oscillations 180 degrees out of phase with the canard and tail motion.
In addition, the main wing is also subjected to a 5 Hz oscillatory twisting
motion whose amplitude decays linearly from 0.5 degrees at the wing tip to
0 degrees at the wing root and takes place about the quarter-chord line. This
composite motion is shown in Figs. 2-5. The BDF2opt scheme is used with 10
subiterations and a physical time step corresponding to 100 steps per cycle
of grid motion.

Figure 2. Simulated aeroelastic motion of F-15 geometry.

Figure 3. Simulated aeroelastic motion of canard geometry.

Figure 4. Simulated aeroelastic motion of wing geometry.

Figure 5. Simulated aeroelastic motion of tail geometry.

The unsteady lift-to-drag ratio (L/D) for the baseline configuration
undergoing the specified motion for 300 time steps is shown as the solid line
in Fig. 6. The L/D behavior begins to exhibit a periodic response after
approximately 100 time steps. The high-frequency oscillations in the profile
are believed to be due to a small unsteadiness in the engine plume shown in
Fig. 7; this behavior is also present when the mesh is held fixed.

Figure 6. Unsteady L/D profile before and after optimization.

Figure 7. Cross-section of unsteady engine plume effects.

The objective function for the current test case is to maximize L/D on the
timestep interval 201-300. The surface grids for the canard, wing, and tail
have been parameterized as shown in Fig. 8, resulting in a set of 98 active
design variables describing the thickness and camber of each surface.
Thinning of the geometry is not permitted.

Figure 8. Spanwise and design variable locations for modified F-15.

Convergence of the objective function is shown in Fig. 9. A large reduction
in the function is obtained after a single design cycle, after which further
improvements are minimal due to many of the design variables having
reached their bound constraints. The final L/D profile is included as the
dashed line in Fig. 6. The resulting shape changes at various spanwise
stations on the canard, wing, and tail are shown in Fig. 10, where the
vertical scale has been exaggerated for clarity. The design procedure has
increased the thickness of the wing and canard, as well as the
camber across all three elements. Closer inspection shows that the trailing
edges of each surface have also been deflected in a downward fashion.

This example has been performed using 128 dual-socket quad-core nodes with 3.0
GHz Intel Xeon processors in a fully-dense fashion for a total of 1,024
computational cores. The wallclock times required for single flow and adjoint
solutions for the current problem are approximately 1 and 1.5 hours, respectively.
For the five design cycles shown in Fig. 9, the optimizer requires ten flow
solutions and five adjoint solutions, or a total wallclock time of approximately
18 hours or 18,400 CPU hours. The disk space necessary to store a single
unsteady flow solution is 136 gigabytes.

Figure 9. Objective function history.

Figure 10. Canard, wing, and tail cross-sections before and after
optimization of modified F-15.